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Chapter 5: Problem 46

Factor completely. $$ a^{9}+b^{12} c^{15} $$

Short Answer

Expert verified
The expression \(a^{9} + b^{12}c^{15}\) cannot be factored further.

Step by step solution

01

Identify the Greatest Common Factor (GCD)

First, check if there is a common factor for the terms in the expression. In this case, the terms have no common numerical or variable factors.
02

Recognize the Sum of Two Terms

Observe that the expression consists of two terms: \(a^{9}\) and \(b^{12} c^{15}\). This is a sum of two terms.
03

Determine Non-Factorable Expression

Check if the sum of these terms can be factored further. Since there are no common factors and it does not fit any standard factoring formulas like the difference of squares or sum of cubes, the expression is already in its simplest form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

greatest common factor
When factoring polynomials, the first step is to look for the Greatest Common Factor (GCF). The GCF is the highest factor that divides all terms in the polynomial without a remainder.
In simpler terms, it鈥檚 the largest number or expression that is shared by all terms. Here鈥檚 how to spot it:

  • Look at the coefficients (the numbers in front of variables).
  • Evaluate each variable and its exponent.
For example, in the exercise, the given terms are \(a^9\) and \(b^{12} c^{15}\). If you check both terms, they do not share any numerical or variable factors. This means there's no common factor to extract from both terms. This indicates that the expression must be checked using other factoring techniques.
sum of terms
The next step is to recognize the structure of the polynomial. Specifically, you need to determine whether it鈥檚 a sum of terms or another form. There are two terms in this case: \(a^9\) and \(b^{12} c^{15}\).
These terms don鈥檛 have common variables or coefficients, indicating that we鈥檙e dealing with a simple sum.

The concept of the 'sum of terms' is important since it rules out certain factoring formulas. If it was a difference, certain methods like difference of squares might apply. But with a sum, the options vary and commonly include:
  • Finding common factors
  • Recognizing standard algebraic identities
In this exercise, it鈥檚 evident there are no common factors or relevant identities, simplifying our work.
non-factorable expression
Sometimes, even after evaluating for common factors and restructuring the expression, you find it doesn't fit any known factoring formulas. These instances result in what's called a 'non-factorable expression'.
That's what's occurring in the given polynomial \(a^9 + b^{12} c^{15}\).
  • Firstly, no common factor was found.
  • Secondly, checking sum-specific factoring methods revealed no fitting formula.

Missing common techniques include:
  • Difference of squares: requires subtraction and squared terms.
  • Sum/Difference of cubes: needs cube terms.
Since \(a^9 + b^{12} c^{15}\) does not match these forms, we classify it as non-factorable, meaning it is already in its simplest form.

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